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A remark on congruence kernels in complemented lattices and pseudocomplemented semilattices. (English) Zbl 0940.06009

Chajda, I. (ed.) et al., Contributions to general algebra 11. Proceedings of the Olomouc workshop ’98 on general algebra, “56. Arbeitstagung Allgemeine Algebra”, Olomouc, Czech Republic, June 12-14, 1998 and of the summer school ’98 on universal algebra and ordered sets, Velk√© Karlovice, Czech Republic, August 31-September 5, 1998. Klagenfurt: Verlag Johannes Heyn. 55-58 (1999).
Let \({\mathfrak L}=(L,\wedge,\vee,0,1)\) be a complemented lattice. Suppose that \(C\subseteq L\) is a congruence class of some \(\theta \in\text{Con} L\). It is proved (Theorem 2.1) that \([0]\theta =\{a\in L\mid c\wedge a=0\) and \(c\vee a\in C\) for some \(c\in C\}\). Thus for every \(\theta\in \text{Con} {\mathfrak L}\) its kernel is determined by every class of \(\theta\). A similar theorem is proved for pseudocomplemented semilattices.
For the entire collection see [Zbl 0914.00059].

MSC:

06C15 Complemented lattices, orthocomplemented lattices and posets
06B10 Lattice ideals, congruence relations
06A12 Semilattices
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